Average Error: 29.7 → 1.0
Time: 2.3m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 211.87865755419:\\ \;\;\;\;(\left(x \cdot x\right) \cdot \left((\frac{1}{3} \cdot x + \frac{-1}{2})_*\right) + 1)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 211.87865755419

    1. Initial program 39.3

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]

    3. Taylor expanded around inf 1.2

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + 1\right) - \frac{1}{2} \cdot {x}^{2}}\]

    4. Simplified1.2

      \[\leadsto \color{blue}{(\left(x \cdot x\right) \cdot \left((\frac{1}{3} \cdot x + \frac{-1}{2})_*\right) + 1)_*}\]

    if 211.87865755419 < x

    1. Initial program 0.1

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied add-exp-log0.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 211.87865755419:\\ \;\;\;\;(\left(x \cdot x\right) \cdot \left((\frac{1}{3} \cdot x + \frac{-1}{2})_*\right) + 1)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))